Problem: What is the slope of the line tangent to $f(x) = x^{2}+x-7$ at $x = 3$ ?
Solution: The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{((x+\Delta x)^{2}+x+\Delta x-7) - (x^{2}+x-7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(x^{2}+2x \Delta x+\Delta x^{2}+x+\Delta x-7) - (x^{2}+x-7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{x^{2}+2(x \Delta x)+\Delta x^{2}+x+\Delta x-7-x^{2}-x+7}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{2(x \Delta x)+\Delta x^{2}+\Delta x}{\Delta x}$ $ = \lim_{\Delta x \to 0} 2x+\Delta x+1$ $ = 2x+1$ $ = (2)(3)+1$ $ = 7$